# Full text of "Treatise On Analysis Vol-Ii"

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```216       XIII    INTEGRATION

for the injection JS?c (X, ju) -» 3? c(X, ju) (with respect to the corresponding
seminorms) because it follows from (13.12.5) that N2(/) ^ N^/) • MX)1/2
for all/e.Sfj?(X,/0.

(13.20.5) In particular (13.9.17), if /j is a bounded (complex) measure, every
function fe%g(X) is 0-integrable, and we have I jfdfj, g HI • ||/|| by
(13.16.5). In other words, / i-> |/ d^ is a continuous linear form on the

Banach space ^c W- But it should be noted that, in general, there exist
continuous linear forms on this space which are not of this type.

(1 3.20.6) The space M£(X) is also the dual of the closure ^g(X) of Jf C(X) in
the Banach space #£(X) (12.15). A function / belonging to *g(X) may be
characterized by the following property :for each s > 0, there exists a compact
subset K ofX such that |/(jc)| ^ sfor allxeX-K. For if/e «?(X), then for
each & > 0 there exists by definition a function g e ^TC(X) such that
||/— #|| :gj e; if K is the support of g, then \f(x)\ ^ e for all x \$ K. Conversely,
suppose that / has the above property, and let h be a continuous mapping
of Xinto [0, 1], with compact support and equal to 1 on K ((3.18.2) and
(4.5.2)); it is clear that \\f-fh\\ £ s and that//ze Jfc(X), hence /e «g(X).
The functions belonging to ^cPO are called (complex-valued) continuous
functions which tend to 0 at infinity. (When X = R, they are indeed the
continuous functions / such that lim /(#) = lim f(x) = 0.) We put

n ^R(X) for the corresponding space of real-valued functions.
When X is compact, we have

PROBLEMS

1.   The space Mi(X) of bounded real measures on a locally compact space X can be
considered as a space of linear forms on each of the following vector spaces :

(1)    the space EI = ^TR(X) of continuous functions with compact support;

(2)    the space E2 = #a(X) of continuous functions which tend to 0 at infinity;

(3)    the space E3 = ^f(X) of bounded continuous functions;

(4)    the space E5 of linear combinations of (upper or lower) semicontinuous
bounded functions;

(5)    the space E6 = ^n(X) of universally measurable bounded functions.

Moreover, if v is any positive measure on X, the space Mi(X, v) of bounded
measures with base v (which may be identified with the space L£(X, v) by virtue of
13.14.4)) can be considered as a space of linear forms on the vector space E4,v of
bounded functions which are continuous almost everywhere with respect to v. y fn is everywhere finite and continuous (because every point
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